A Comment on the Relationship between the Modal and Time-Distance Formulations of Local Helioseismology
The relationship between the time-distance and modal-decomposition approaches to solar active region seismology is clarified through the consideration of the oscillations of a plane-parallel, isentropic polytrope. It is demonstrated by direct construction that a wave packet formed through the superposition of neighboring p-modes interferes constructively along a ray bundle that follows the appropriate WKBJ ray path obtained by using the eikonal approximation. Because the actual power envelope of the solar 5 minute oscillations restricts the excited p-modes to rather low radial orders, the ray bundles are diffuse and sample portions of the solar envelope that are some ~10-30 Mm distant from the nominal WKBJ ray path. This behavior is consistent with the fact that the eikonal approximation becomes valid only in the limiting case of large radial orders (n >> 1). The p-mode wave packets that are isolated by employing the time-distance methods must therefore be described either as a superposition of individual p-modes (a wave packet), or as a sum of ray paths (a ray bundle), depending upon which representation proves to be optimal for the given circumstances.